Theorem recexprlemm 7991

Description: B is inhabited. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlemm	|- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

Distinct variable groups:   r, q, x, y, A   B, q, r, x, y

Proof of Theorem recexprlemm

Step	Hyp	Ref	Expression
1		prop 7842	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7845	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
3		recclnq 7759	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) e. Q. )
4		nsmallnqq 7779	. . . . . . 7 |- ( ( *Q ` x ) e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
5	3, 4	syl 14	. . . . . 6 |- ( x e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
6	5	adantr 276	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q <Q ( *Q ` x ) )
7		recrecnq 7761	. . . . . . . . . . . 12 |- ( x e. Q. -> ( *Q ` ( *Q ` x ) ) = x )
8	7	eleq1d 2307	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) <-> x e. ( 2nd ` A ) ) )
9	8	anbi2d 468	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) ) )
10		breq2 4134	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( q <Q y <-> q <Q ( *Q ` x ) ) )
11		fveq2 5695	. . . . . . . . . . . . . 14 |- ( y = ( *Q ` x ) -> ( *Q ` y ) = ( *Q ` ( *Q ` x ) ) )
12	11	eleq1d 2307	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) )
13	10, 12	anbi12d 477	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) ) )
14	13	spcegv 2913	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
15	3, 14	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
16	9, 15	sylbird 170	. . . . . . . . 9 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
17		recexpr.1	. . . . . . . . . 10 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
18	17	recexprlemell 7989	. . . . . . . . 9 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
19	16, 18	imbitrrdi 162	. . . . . . . 8 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) ) )
20	19	expcomd 1491	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 2nd ` A ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) ) )
21	20	imp 124	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) )
22	21	reximdv 2651	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( E. q e. Q. q <Q ( *Q ` x ) -> E. q e. Q. q e. ( 1st ` B ) ) )
23	6, 22	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q e. ( 1st ` B ) )
24	23	rexlimiva 2663	. . 3 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. q e. Q. q e. ( 1st ` B ) )
25	1, 2, 24	3syl 17	. 2 |- ( A e. P. -> E. q e. Q. q e. ( 1st ` B ) )
26		prml 7844	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 1st ` A ) )
27		1nq 7733	. . . . . . . 8 |- 1Q e. Q.
28		addclnq 7742	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
29	3, 27, 28	sylancl 417	. . . . . . 7 |- ( x e. Q. -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
30		ltaddnq 7774	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
31	3, 27, 30	sylancl 417	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
32		breq2 4134	. . . . . . . 8 |- ( r = ( ( *Q ` x ) +Q 1Q ) -> ( ( *Q ` x ) <Q r <-> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) )
33	32	rspcev 2929	. . . . . . 7 |- ( ( ( ( *Q ` x ) +Q 1Q ) e. Q. /\ ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
34	29, 31, 33	syl2anc 415	. . . . . 6 |- ( x e. Q. -> E. r e. Q. ( *Q ` x ) <Q r )
35	34	adantr 276	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
36	7	eleq1d 2307	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) <-> x e. ( 1st ` A ) ) )
37	36	anbi2d 468	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) ) )
38		breq1 4133	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( y <Q r <-> ( *Q ` x ) <Q r ) )
39	11	eleq1d 2307	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) )
40	38, 39	anbi12d 477	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) ) )
41	40	spcegv 2913	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
42	3, 41	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
43	37, 42	sylbird 170	. . . . . . . . 9 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
44	17	recexprlemelu 7990	. . . . . . . . 9 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
45	43, 44	imbitrrdi 162	. . . . . . . 8 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) ) )
46	45	expcomd 1491	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 1st ` A ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) ) )
47	46	imp 124	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) )
48	47	reximdv 2651	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( E. r e. Q. ( *Q ` x ) <Q r -> E. r e. Q. r e. ( 2nd ` B ) ) )
49	35, 48	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. r e. ( 2nd ` B ) )
50	49	rexlimiva 2663	. . 3 |- ( E. x e. Q. x e. ( 1st ` A ) -> E. r e. Q. r e. ( 2nd ` B ) )
51	1, 26, 50	3syl 17	. 2 |- ( A e. P. -> E. r e. Q. r e. ( 2nd ` B ) )
52	25, 51	jca 306	1 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   E.wrex 2529   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   1Qc1q 7648   +Q cplq 7649   *Qcrq 7651   <Q cltq 7652   P.cnp 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-inp 7833

This theorem is used by:  recexprlempr  7999